Nordhaus-Gaddum-type theorems for maximum average degree

TL;DR

This work advances Nordhaus–Gaddum–type results for the maximum average degree Mad by studying $M(k,n)$, the maximum possible sum of Mad over $k$-edge-decompositions of the complete graph $K_n$. It develops tight general bounds, exact values for small $k$ (notably $M(2,n)$), and precise descriptions for the corresponding list-variant $M^{\mathsf{L}}(k,N)$, linking these extremal values to Steiner systems and graph designs. The authors introduce a structural reformulation of optimal decompositions, establish a continuous-relaxation upper bound $M(k,n)<\sqrt{k}\,n$, and use Steiner designs, finite geometries, and Wilson-type decompositions to yield sharp or nearly sharp $M(k,n)$ in broad regimes, including asymptotic growth $M(k,n)\sim (1-o_k(1))\sqrt{k}\,n$. They also connect these results to other invariants such as clique number, chromatic and choice numbers, and degeneracy, showing that under certain conditions these invariants inherit the same extremal behavior. Overall, the paper provides a comprehensive framework for Mad-Nordhaus–Gaddum-type extremals with broad combinatorial constructions and significant implications for related graph parameters.

Abstract

A $k$-decomposition $(G_1,\dots,G_k)$ of a graph $G$ is a partition of its edge set into $k$ spanning subgraphs $G_1,\dots,G_k$. The classical theorem of Nordhaus and Gaddum bounds $χ(G_1) + χ(G_2)$ and $χ(G_1) χ(G_2)$ over all 2-decompositions of $K_n$. For a graph parameter $p$, let $p(k,G) = \max \{ \sum_{i=1}^k p(Gi) \}$, taken over all $k$-decompositions of graph $G$. In this paper we consider $M(k,K_n) = M(k,n) = \max \{ \sum_{i=1}^k \mathrm{Mad}(G_i) \}$, taken over all $k$-decompositions of the complete graph $K_n$, where $\mathrm{Mad}(G)$ denotes the maximum average degree of $G$, $\mathrm{Mad}(G) = \max \{ 2e(H)/|H| : H \subseteq G \} = \max \{d(H) : H \subseteq G \}$. Among the many results obtained in this paper we mention the following selected ones. (1) $M(k, n) < \sqrt{k} n$, and $\lim_{k\to\infty} ( \liminf_{n\to\infty} \frac{M(k,n)}{\sqrt{k}\,n} ) = 1$. (2) Exact determination of $M(2,n)$. (3) Exact determination of $M(k,n)$ when $k = \binom{n}{2} - t$, $0 \leq t\leq (n-1)^2/3$. Applications of these bounds to other parameters considered before in the literature are given.

Theorems & Definitions (74)

• Definition 1
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof
• Theorem 8